数学第7题
积分范围为 1/4 圆:x² + y² =1 在第 1 象尺茄限。换成极坐标,则 r =[0, 1],θ = [0, π/2]
那么,这个积分就可以变换为:
=∫∫√[(1-r²)/(1+r²)] * r * dr*dθ
=∫dθ∫√[(1-r²)/(1+r²)] * r*dr
=π/2 * ∫√[(1-r²)/(1+r²)] * 1/2 * d(r²)
=π/4 * ∫陵脊察√[(1-u)/(1+u)] * du 注:u = r²,u 的积分范围 = [0, 1]
=π/4 * ∫√(1-u²) /(1+u) * du
=π/4 * ∫cosα * d(sinα)/(1+sinα) 注:设野差 u = sinα,则 du = d(sinα),α = [0, π/2]
=π/4 * ∫cos²α * dα/(1+sinα)
=π/4 * ∫(1-sin²α)*dα/(1+sinα)
=π/4 * ∫(1-sinα)*dα
=π/4 * [∫dα - ∫sinα * dα]
=π/4 * [α + cosα]|α=0→π/2
=π/4 * [(π/2 -0) + cos(π/2) - cos0]
=π/4 * (π/2 - 0 - 1)
=π²/8 - π/4
条件不全,无法计算。